# American Institute of Mathematical Sciences

January  2012, 17(1): 153-171. doi: 10.3934/dcdsb.2012.17.153

## Stability of an efficient Navier-Stokes solver with Navier boundary condition

 1 School of Science, East China University of Science and Technology, Shanghai, 200237, China 2 Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

Received  March 2011 Revised  July 2011 Published  October 2011

In this paper, we study the stability of an efficient numerical scheme based on pressure separation for the Navier-Stokes equations with Navier slip boundary condition in a bounded domain with smooth boundary. The method was introduced in [7, 8] for the Navier-Stokes equation with no-slip boundary condition which decouples the updates of pressure and velocity through explicit time stepping for pressure. The scheme was shown to be very efficient and unconditionally stable. In this paper, we extend this pressure separation method to the problem with Navier slip boundary condition and prove the unconditional stability of the resulting numerical scheme under certain condition on the curvature of the boundary.
Citation: Jie Liao, Xiao-Ping Wang. Stability of an efficient Navier-Stokes solver with Navier boundary condition. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 153-171. doi: 10.3934/dcdsb.2012.17.153
##### References:
 [1] D. Einezl, P. Panzer and M. Liu, Boundary condition for fluid flow: Curved or rough surfaces, Physical Review Letters, 64 (1990), 2269-2272. doi: 10.1103/PhysRevLett.64.2269. [2] L. C. Evans, "Partial Differential Equations," 2nd edition, Graduate Studies in Mathematics, 19, Amer. Math. Soc., Providence, RI, 2010. [3] C. Foias, O. Manley, R. Rosa and R. Temam, "Navier-Stokes Equations and Turbulence," Encyclopedia of Mathematics and its Applications, 83, Cambridge University Press, 2001. [4] J. L. Guermond, P. Minev and J. Shen, An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Engrg., 195 (2006), 6011-6045. doi: 10.1016/j.cma.2005.10.010. [5] Q.-L. He and X.-P. Wang, Numerical study of the effect of Navier slip on the driven cavity flow, Z. Angew. Math. Mech., 89 (2009), 857-868. doi: 10.1002/zamm.200900245. [6] O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," 2nd English edition, revised and enlarged, Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969. [7] J.-G. Liu, J. Liu and R. L. Pego, Divorcing pressure from viscosity in incompressible Navier-Stokes dynamics,, preprint, (). [8] J.-G. Liu, Jie Liu and R. Pego, Stability and convergence of efficient Navier-Stokes solvers via a commutator estimate, Comm. Pure Appl. Math., 60 (2007), 1443-1487. doi: 10.1002/cpa.20178. [9] C. L. M. H. Navier, Memoire sur les lois du mouvement des fluides, Mem. Acad. R. Sci. Inst. Fr., 6 (1823), 389-440. [10] C. Neto, D. R. Evans, E. Bonaccurso, H.-J. Butt and V. S. J. Craig, Boundary slip in Newtonian liquids: A review of experimental studies, Rep. Prog. Phys., 68 (2005), 2859-2897. doi: 10.1088/0034-4885/68/12/R05. [11] T.-Z. Qian and X.-P. Wang, Driven cavity flow: From molecular dynamics to continuum hydrodynamics, Multiscale Model. Simul., 3 (2005), 749-763. doi: 10.1137/040604868.

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##### References:
 [1] D. Einezl, P. Panzer and M. Liu, Boundary condition for fluid flow: Curved or rough surfaces, Physical Review Letters, 64 (1990), 2269-2272. doi: 10.1103/PhysRevLett.64.2269. [2] L. C. Evans, "Partial Differential Equations," 2nd edition, Graduate Studies in Mathematics, 19, Amer. Math. Soc., Providence, RI, 2010. [3] C. Foias, O. Manley, R. Rosa and R. Temam, "Navier-Stokes Equations and Turbulence," Encyclopedia of Mathematics and its Applications, 83, Cambridge University Press, 2001. [4] J. L. Guermond, P. Minev and J. Shen, An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Engrg., 195 (2006), 6011-6045. doi: 10.1016/j.cma.2005.10.010. [5] Q.-L. He and X.-P. Wang, Numerical study of the effect of Navier slip on the driven cavity flow, Z. Angew. Math. Mech., 89 (2009), 857-868. doi: 10.1002/zamm.200900245. [6] O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," 2nd English edition, revised and enlarged, Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969. [7] J.-G. Liu, J. Liu and R. L. Pego, Divorcing pressure from viscosity in incompressible Navier-Stokes dynamics,, preprint, (). [8] J.-G. Liu, Jie Liu and R. Pego, Stability and convergence of efficient Navier-Stokes solvers via a commutator estimate, Comm. Pure Appl. Math., 60 (2007), 1443-1487. doi: 10.1002/cpa.20178. [9] C. L. M. H. Navier, Memoire sur les lois du mouvement des fluides, Mem. Acad. R. Sci. Inst. Fr., 6 (1823), 389-440. [10] C. Neto, D. R. Evans, E. Bonaccurso, H.-J. Butt and V. S. J. Craig, Boundary slip in Newtonian liquids: A review of experimental studies, Rep. Prog. Phys., 68 (2005), 2859-2897. doi: 10.1088/0034-4885/68/12/R05. [11] T.-Z. Qian and X.-P. Wang, Driven cavity flow: From molecular dynamics to continuum hydrodynamics, Multiscale Model. Simul., 3 (2005), 749-763. doi: 10.1137/040604868.
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